The Geometry of the Schwarzschild Metric

Previously, we derived the Schwarzschild metric by solving Einstein's field equations under the assumptions of spherical symmetry, staticity, and asymptotic flatness. The resulting metric describes the spacetime geometry outside a non-rotating, uncharged, spherically symmetric mass. The Schwarzschild metric is given by

where is the Schwarzschild radius. Here, we will explore the geometric properties of the Schwarzschild metric and how it describes the curvature of spacetime around a massive object.

Table of Contents

• Table of Contents
• Gravitational Time Dilation
  • Outside the Schwarzschild Radius: r > rₛ, gₜₜ in [0, 1)

Gravitational Time Dilation

Time dilation refers to the difference in the time coordinate of two coordinate systems. As we are dealing with spherically symmetric spacetimes, we just need to consider the time and radial components of the metric, and . Our metric is thus two-dimensional, and we can write it as

Let there be a mass at the origin (i.e. ) which flows along the coordinate. We will draw gridlines, but remember that these do not represent physical distance! We never rely on our eyes or the coordinates to determine physical distances in curved spacetimes. We only rely on the metric to determine angles and distances.

First, notice the length of , as given by the inner product with itself:

Four cases arise:

1. First, as , we have , which is the same as in flat spacetime.
2. Second, as from the right, becomes smaller and smaller, but remains positive (within ). This means that outside the Schwarzschild radius, is a timelike vector, but its length shrinks as we approach .
3. At , becomes a null/lightlike vector, as . This means that at the Schwarzschild radius, is neither timelike nor spacelike.
4. Finally, as , , meaning that is a spacelike vector inside the Schwarzschild radius. This means that inside the Schwarzschild radius, the coordinate behaves like a spatial coordinate. Observers can traverse the coordinate in both directions like a spatial coordinate.

We will discuss each of these cases in turn.

Outside the Schwarzschild Radius: r > rₛ, gₜₜ in [0, 1)

Remember that the coordinate is responsible for measuring time. As such, a difference in length of across regions of spacetime indicates a difference in the flow of time. To calculate the flow of time, we can use the metric to compute the proper time along a worldline:

Consider a scenario in which two observers are located at different but constant radial distances from the mass . One observer is at a distance from the mass, while the other is at a much larger distance, effectively at infinity.

Note that this is not a realistic scenario. The observers would naturally freefall towards the mass due to gravity. As such, they will require some form of propulsion to maintain their positions, or be standing on the surface of a planet or star, held up by electromagnetic forces. However, this is a useful thought experiment to understand gravitational time dilation.

Anyways, we have

In a more familiar form, we have

This is similar to the time dilation formula in special relativity, where the velocity corresponds to , which is the escape velocity from a mass at a distance .

As from the right, , meaning that the conversion factor between and becomes very small. This means that time slows down as we approach the Schwarzschild radius from the outside. To take an example, suppose we are at . Then, we have

This means that for every second that passes for a distant observer, only about 0.577 seconds pass for an observer at . If a year passes for a distant observer, only about 211 days pass for the observer at . This phenomenon is known as gravitational time dilation, named because it is a form of time dilation specifically caused by gravity.

In the same way we can compute the proper time, we can also consider the proper length between two radial positions and . It's the same integral, but the curve needs to be spacelike (i.e. the inner product of the tangent vector with itself needs to be negative).

This time, we have that is constant, so the only non-zero component of the tangent vector is .

Unlike the previous integral, we can't just pull out the radical, as it depends on . Let's look at the infinitesimal case:

In other words, a small change in the radial coordinate corresponds to a larger proper length , as for all . As , , meaning that far away from the mass, the proper length is approximately equal to the coordinate length. Conversely, as from the right, , meaning that near the Schwarzschild radius, a small change in the radial coordinate corresponds to a very large proper length.

There is an analytical solution to the integral, which yields